
with the initial condition \(P_u({\bf x}^{\prime },t|{\bf x},t)= \delta ({\bf x}-{\bf x}^{\prime })\)  (Appendix ). This conditional probability density can be used directly to compute the cost-to-go function \(J({\bf x},t)\) , and consequently the optimal control \({\bf u}^*\) , as discussed in detail in Appendix . Interestingly, the evolution of the optimally controlled conditional distribution \(P_u({\bf x}^{\prime },t^{\prime }|{\bf x},t)\)  resembles the natural evolutionary dynamics (eq. REF  with \({\bf u}=0\) ) with an extra annihilation term \( V({\bf x},t)/\lambda \) ; see Appendix  and ref. [1]}. Therefore, artificial selection acts as an importance sampling protocol over each selection cycle (e.g. each generation) that removes (annihilates) individuals from the population with a rate proportional to their distance from the evolutionary target \(\sim V(|{\bf x}_t-{\bf x}^*|)/\lambda \) ; see Fig. REF A. Specifically, at each time point, this protocol generates a phenotypic distribution consistent with the evolutionary process under optimal artificial selection in eq. REF  (Fig. REF B,C), without an explicit knowledge of the selection protocol \({\bf u}^*({\bf x},t)\) ; see Appendix  and refs. [1]}. This result is highly practical for complex evolutionary processes, for which an analytical description of the optimal control protocol is inaccessible.
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